A geometric characterization of the upper bound for the span of the jones polynomial

Gonzalez Meneses, Juan and Gonzalez Manchon, Pedro Maria (2011). A geometric characterization of the upper bound for the span of the jones polynomial. "Journal of Knot Theory and Its Ramifications", v. 20 (n. 7); pp. 1059-1071. ISSN 0218-2165. https://doi.org/10.1142/S0218216511009005.

Description

Title: A geometric characterization of the upper bound for the span of the jones polynomial
Author/s:
  • Gonzalez Meneses, Juan
  • Gonzalez Manchon, Pedro Maria
Item Type: Article
Título de Revista/Publicación: Journal of Knot Theory and Its Ramifications
Date: July 2011
ISSN: 0218-2165
Volume: 20
Subjects:
Faculty: E.U.I.T. Industrial (UPM)
Department: Matemática Aplicada
Creative Commons Licenses: Recognition - No derivative works - Non commercial

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Abstract

Let D be a link diagram with n crossings, sA and sB be its extreme states and |sAD| (respectively, |sBD|) be the number of simple closed curves that appear when smoothing D according to sA (respectively, sB). We give a general formula for the sum |sAD| + |sBD| for a k-almost alternating diagram D, for any k, characterizing this sum as the number of faces in an appropriate triangulation of an appropriate surface with boundary. When D is dealternator connected, the triangulation is especially simple, yielding |sAD| + |sBD| = n + 2 - 2k. This gives a simple geometric proof of the upper bound of the span of the Jones polynomial for dealternator connected diagrams, a result first obtained by Zhu [On Kauffman brackets, J. Knot Theory Ramifications6(1) (1997) 125–148.]. Another upper bound of the span of the Jones polynomial for dealternator connected and dealternator reduced diagrams, discovered historically first by Adams et al. [Almost alternating links, Topology Appl.46(2) (1992) 151–165.], is obtained as a corollary. As a new application, we prove that the Turaev genus is equal to the number k of dealternator crossings for any dealternator connected diagram

More information

Item ID: 14153
DC Identifier: http://oa.upm.es/14153/
OAI Identifier: oai:oa.upm.es:14153
DOI: 10.1142/S0218216511009005
Official URL: http://www.worldscientific.com/doi/abs/10.1142/S0218216511009005?journalCode=jktr
Deposited by: Memoria Investigacion
Deposited on: 19 Dec 2012 10:58
Last Modified: 21 Apr 2016 13:40
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